Elsevier

Information Sciences

Volume 176, Issue 7, 6 April 2006, Pages 759-771
Information Sciences

Longest fault-free paths in hypercubes with vertex faults

https://doi.org/10.1016/j.ins.2005.01.011Get rights and content

Abstract

The hypercube is one of the most versatile and efficient interconnection networks (networks for short) so far discovered for parallel computation. Let f denote the number of faulty vertices in an n-cube. This study demonstrates that when f  n  2, the n-cube contains a fault-free path with length at least 2n  2f  1 (or 2n  2f  2) between two arbitrary vertices of odd (or even) distance. Since an n-cube is a bipartite graph with two partite sets of equal size, the path is longest in the worst-case. Furthermore, since the connectivity of an n-cube is n, the n-cube cannot tolerate n  1 faulty vertices. Hence, our result is optimal.

Introduction

The hypercube is one of the most versatile and efficient interconnection networks (networks for short) discovered to date for parallel computation. The hypercube is ideally suited to both special-purpose and general-purpose tasks and can efficiently simulate many other networks of the same size [11]. An embedding of one guest graph G into another host graph H is a one-to-one mapping m from the vertex set of G to the vertex set of H [11]. An edge of G corresponds to a path of H under m. Embedding has been the subject of intensive study, with the hypercube being the host graph and various graphs being the guest graph.

Linear arrays and rings, which are two of the most fundamental networks for parallel and distributed computation, are suitable for designing simple algorithms with low communication costs. Numerous efficient algorithms designed on linear arrays and rings for solving various algebraic problems and graph problems can be found in [1], [11]. Linear arrays and rings can also be used as control/data flow structures for distributed computation in arbitrary networks. An application of longest paths to a practical problem was encountered in the on-line optimization of a complex Flexible Manufacturing System (see [2]). These applications motivate the embedding of paths and cycles in networks. Since processor or link faults may develop in real world networks, it is important to consider faulty networks. The problems of diameter [4], routing [6], multicasting [13], broadcasting [16], gossiping [5], and embedding [7], [12], [17] have been solved in various faulty networks. This study considers the problem of fault-tolerant embedding. Throughout this study, a number of terms—network and graph, processor and vertex, and link and edge—are used interchangeably.

Previously, the problem of fault-tolerant embedding on the hypercube has been studied in [3], [8], [10], [14], [15]. The n dimensional hypercube (n-cube) with f faulty links contains a fault-free ring of length 2n, where 1  f  n  2[10]. A fault-free cycle with length at least 2n  2f can be embedded in an n-cube with f faulty vertices, where 1  f  2n   4 [8]. For an n-cube with fe  n  4 faulty links and fv  n  1 faulty vertices such that fe + fv  n  1, a fault-free cycle with length at least 2n  2fv can be obtained [15]. In an n-cube with f faulty links, a fault-free path of length 2n  1 (or 2n  2) exists between two arbitrary vertices of the odd (or even) distance, where 1  f  n  2 [14]. For any n-cube (n  3) with ⩽2n  5 link faults in which each vertex is incident to at least two nonfaulty links, a fault-free ring of length 2n can be obtained [3].

This study only considers the problem of embedding a longest fault-free path in an n-cube with faulty vertices. It is demonstrated that in an n-cube with f faulty vertices, there exists a fault-free path of length 2n  2f  1 (or 2n  2f  2) between two arbitrary vertices of odd (or even) distance, where 1  f  n   2. Since an n-cube is a bipartite graph with two partite sets of equal size, the path is longest in the worst-case. Furthermore, since the connectivity of an n-cube is n, the n-cube cannot tolerate n  1 faulty vertices. Thus, or result is optimal.

Section snippets

Preliminaries

An n-cube is an undirected graph with 2n vertices, each labeled with a distinct binary string b1b2bn. Vertices b1bibn and b1bi¯bn are joined by an edge along dimension i, where 1  i  n and bi¯ represents the one complement of bi. Moreover, suppose X = x1x2xn and Y = y1y2yn. In the rest of the paper, X(i) is used to denote the binary string x1xi¯xn, and dH(X, Y) is used to denote the Hamming distance between X and Y, namely, the number of different bits between X and Y. A path from X to Y is

Longest fault-free paths with vertex faults

This section embeds a longest path between two distinct vertices X and Y in an n-cube. The basic embedding idea uses a recursive construction. It is assumed that a longest path between two arbitrary vertices can be embedded in an (n  1)-cube. An n-cube can be partitioned into two vertex-disjoint (n  1)-cubes, and X and Y can be located in different (n  1)-cubes. Two adjacent healthy vertices, Z and Z′, will be chosen such that Z (or Z′) and X (or Y) are in the same (n  1)-cube. The longest XZ (or Y

Discussion and conclusion

Fault tolerance is an important research subject in the area of the multi-process computer systems, and many studies have focused on the vertex-fault tolerant or edge-fault tolerant properties of some specific networks. The induction proof is a simple and powerful scheme that is especially suited to recursive networks. Using induction proofs, this study has shown that an n-cube with f  n  2 faulty vertices contains a fault-free path with length at least 2n  2f  1 (or 2n  2f  2) between two arbitrary

Acknowledgement

The author would like to express his deepest gratitude to the anonymous referees for their insightful comments. They have improved very much the readability of the paper. The author also thanks the National Science Council of the Republic of China, Taiwan, for financially supporting this research under Contract No. NSC 92-2213-E-239-010-.

References (17)

There are more references available in the full text version of this article.

Cited by (66)

  • Hamiltonian paths in hypercubes with local traps

    2017, Information Sciences
    Citation Excerpt :

    Hamiltonian paths and cycles in hypercubes with faulty edges or vertices have been investigated by many authors; see [1,6,18,21–26,29,30]. Other problems that have attracted considerable attention in the literature in recent years include: the existence of long paths in hypercubes [7,16], the paired many-to-many disjoint path covers in hypercubes [3–5,9,15], and Hamiltonian paths and cycles in some other variants of hypercube (balanced, folded, augmented, k-ary, crossed or locally twisted hypercubes) [8,10–13,17,31]. [25]

  • Two node-disjoint paths in balanced hypercubes

    2014, Applied Mathematics and Computation
View all citing articles on Scopus
View full text